Ranking sets of objects using the Shapley valueand other regular semivalues
Stefano Moretti, Alexis Tsoukias
Laboratoire d’Analyse et Modelisation de Systemes pour l’Aide a la DEcision(Lamsade)
CNRS UMR7243, Paris Dauphine University
DIMACS Workshop on Algorithmic Aspects of InformationFusion (WAIF), November 8 - 9, 2012 DIMACS Center,
Rutgers University
Summary
Preferences over sets
Properties that prevent the interaction
Alignment with regular semivalues
Interaction among objects
Two recent papers
- Moretti S., Tsoukias A. (2012). Ranking Sets of PossiblyInteracting Objects Using Shapley Extensions. In ThirteenthInternational Conference on the Principles of KnowledgeRepresentation and Reasoning (KR2012).
- Lucchetti R., Moretti S., Patrone F. (2012) A probabilisticapproach to ranking sets of interacting objects, in progress.
Central question
How to derive a ranking over the set of all subsets of N in a waythat is “compatible” with a primitive ranking over the single
elements of N?
- Relevant number of papers focused on the problem of deriving apreference relation on the power set of N from a preferencerelation over single objects in N. Most of them provide anaxiomatic approach (Kannai and Peleg (1984), Barbera et al(2004), Bossert (1995), Fishburn (1992), Roth (1985) etc.)
- Extension axiom: Given a total preorder < on N, we say that atotal preorder w on 2N is an extension of < if and only if for eachx , y ∈ N,
{x} w {y} ⇔ x < y
Well-known properties prevent interaction
Axiom [Responsiveness, RESP] A total preorder w on 2N satisfiesthe responsiveness property iff for all A ∈ 2N \ {N, ∅}, for all x ∈ Aand for all y ∈ N \ A the following conditions holds
A w (A \ {x}) ∪ {y} ⇔ {x} w {y}
- This axiom was introduced by Roth (1985) studying colleges’preferences for the “college admission problem” (see also Gale andShapley (1962)).
- Bossert (1995) used the same property for ranking sets ofalternatives with a fixed cardinality and to characterize the class ofrank-ordered lexicographic extensions.
Well-known extensions prevent interaction
Most of the axiomatic approaches from the literature make use ofthe RESP axiom to prevent any kind of interaction among theobjects in N.:
- max and min extensions (Kreps 1979, Barbera, Bossert, andPattanaik 2004)
- lexi-min and lexi-max extensions (Holzman 1984, Pattanaik andPeleg 1984)
- median-based extensions (Nitzan and Pattanaik 1984)
- rank-ordered lexicographic extensions (Bossert 1995)
- many others...
Basic-Basic on coalitional games
A coalitional game (many names...) is a pair (N, v), where Ndenotes the finite set of players and v : 2N → R is thecharacteristic function, with v(∅) = 0.
Given a game, a regular semivalue (see Dubey et al. 1981,Carreras and Freixas 1999; 2000) may be computed to convertinformation about the worth that coalitions can achieve into apersonal attribution (of payoff) to each of the players:
πpi (v) =∑
S⊂N:i /∈S
ps
(v(S ∪ {i})− v(S)
)for each i ∈ N, where ps represents the probability that a coalitionS ∈ 2N (of cardinality s) with i /∈ S forms. So coalitions of thesame size have the same probability to form!
(of course∑n−1
s=0
(n−1s
)ps = 1, but we also assume ps > 0.)
Shapley and Banzhaf regular semivalues
- The Shapley value (Shapley 1953) is a regular semivalue πp(v),where
ps =1
n(n−1
s
) =s!(n − s − 1)!
n!
for each s = 0, 1, . . . , n− 1 (i.e., the cardinality is selected with thesame probability).
- Another very well studied probabilistic value is the Banzhaf value(Banzhaf III 1964), which is defined as the regular semivalueπp(v), where
ps =1
2n−1
for each s = 0, 1, . . . , n − 1, (i.e., each coalition has an equalprobability to be chosen)
πp-aligned total preorders
Given a total preorder w on 2N , we denote by V (w) the class ofcoalitional games that numerically represent w (for eachS ,V ∈ 2N , S w V ⇔ u(S) ≥ u(V ) for each u ∈ V (w)).
DEF. Let πp be a regular semivalue. A total prorder w on 2N isπp-aligned iff for each numerical representation v ∈ V (w) we havethat
{i} w {j} ⇔ πpi (v) ≥ πpj (v)
for all i , j ∈ N.
Here we use regular semivalues to impose a constraint to thepossibilities of interaction among objects: complementarities orredundancy are possible but, globally, their effects cannotoverwhelm the limitation imposed by the original ranking.
Example: Shapley-aligned total preorder...
For each coalitional game v , the Shapley value is denoted byφ(v) = πp(v).Let N = {1, 2, 3} and let wa be a total preorder on N such that{1, 2, 3} Aa {3} Aa {2} Aa {1, 3} Aa {2, 3} Aa {1} Aa {1, 2} Aa
∅.
For every v ∈ V (wa)
φ2(v)− φ1(v) =1
2
(v(2)− v(1)
)+
1
2
(v(2, 3)− v(1, 3)
)> 0
On the other hand
φ3(v)− φ2(v) =1
2
(v(3)− v(2)
)+
1
2
(v(1, 3)− v(1, 2)
)> 0.
... πp-aligned for other regular semivalues
Note that wa is πp-aligned for every regular semivalue such thatp0 ≥ p2:
πp2(v)−πp1(v) = (p0+p1)(v(2)−v(1)
)+(p1+p2)
(v(2, 3)−v(1, 3)
)> 0
On the other hand
πp3(v)−πp2(v) = (p0+p1)(v(3)−v(2)
)+(p1+p2)
(v(1, 3)−v(1, 2)
)> 0
for every v ∈ V (wa).
Total preorder πp-aligned for no regular semivalues
It is quite possible that for a given preorder there is no πp-ordinalsemivalue associated to it. It is enough, for instance, to considerthe case N = {1, 2, 3} and the following total preorder:
N A {1, 2} A {2, 3} A {1} A {1, 3} A {2} A {3} A ∅.
Then it is easy to see that 1 and 2 cannot be ordered since, fixed asemivalue p the quantity
πp2(v)−πp1(v) = (p0+p1)(v({1})−v({2}))+(p1+p2)(v({1, 3})−v({2, 3}))
can be made both positive and negative by suitable choices of v .
Proposition Let w be a total preorder on 2N . If w satisfies theRESP property, then it is πp-aligned with every regular semivalueπp.
- All the extensions from the literature listed in the previous slideare πp-aligned with all regular semivalues...
{1, 2, 3} Aa {3} Aa {2} Aa {1, 3} Aa {2, 3} Aa {1} Aa {1, 2} Aa ∅is not RESP but is πp-aligned with all πp such that p0 ≥ p2.
- We can say something more....
Monotonic total preorders
Axiom [Monotonicity, MON] A total preorder w on 2N satisfiesthe monotonicity property iff for each S ,T ∈ 2N we have that
S ⊆ T ⇒ T w S .
wa introduced in the previous example does not satisfy MON:{1, 2, 3} Aa {3} Aa {2} Aa {1, 3} Aa {2, 3} Aa {1} Aa {1, 2} Aa
∅.
- Min extension is a πp-aligned for all regular semivalues, itsatisfies RESP, but it does not satisfy MON.
An axiomatic characterization (with no interaction)
Let w be a total preorder on 2N . For each S ∈ 2N \ {∅}, denote bywS the restriction of w on 2S such that for each U,V ∈ 2S ,
U w V ⇔ U wS V .
Theorem Let πp be a regular semivalue. Let w be a total preorderon 2N which satisfies the MON property. The following twostatements are equivalent:
(i) w satisfies the RESP property.(ii) wS is πp-aligned for every S ∈ 2N \ {∅}.
- side-product: for a large family of coalitional games all regularsemivalues are ordinal equivalent (e.g. airport games (Littlechildand Owen (1973), Littlechild and Thompson (1977))
A generalization of RESP which admits the interaction
We denote by Σsij the set of all subsets of N of cardinality s which
do not contain neither i nor j , i.e.Σsij = {S ∈ 2N : i , j /∈ S , |S | = s}.
Order the sets S1,S2, . . . ,Sns in Σsij when you add i and j ,
respectively:S1 ∪ {i} Sl(1) ∪ {j}|⊔
|⊔
S2 ∪ {i} Sl(2) ∪ {j}|⊔
|⊔
. . . . . .|⊔
|⊔
Sns ∪ {i} Sl(ns) ∪ {j}
Axiom[Permutational Responsiveness, PR]
We denote by Σsij the set of all subsets of N of cardinality s which
do not contain neither i nor j , i.e.Σsij = {S ∈ 2N : i , j /∈ S , |S | = s}.
Order the sets S1,S2, . . . ,Sns in Σsij when you add i and j ,
respectively:S1 ∪ {i} w Sl(1) ∪ {j}|⊔
|⊔
S2 ∪ {i} w Sl(2) ∪ {j}|⊔
|⊔
. . . w . . .|⊔
|⊔
Sns ∪ {i} w Sl(ns) ∪ {j}
⇔ {i} w {j}
Again a sufficient condition...
Proposition Let w be a total preorder on 2N . If w satisfies the PRproperty, then w is πp-aligned with every regular semivalue.
- Consider the (Shapley-aligned) total prorder wa of previous{1, 2, 3} Aa {3} Aa {2} Aa {1, 3} Aa {2, 3} Aa {1} Aa {1, 2} Aa
∅. Note that {2} A {1}, but {1, 3} A {2, 3}.
- {1, 2, 3, 4} Ab {2, 3, 4} Ab {3, 4} Ab {4} Ab {3} Ab {2} Ab
{2, 4} Ab {1, 4} Ab {1, 3} Ab {2, 3} Ab {1, 3, 4} Ab {1, 2, 4} Ab
{1, 2, 3} Ab {1, 2} Ab {1} Ab ∅ is πp-aligned for all p but does notsatisfy the PR property.
Work in progress: Lucchetti, Moretti, Patrone (2012) Aprobabilistic approach to ranking sets of interacting objects
- A new interpretation of πp-aligned total preorders in terms of“ranking sets of objects” under uncertainty.
- Characterizations of total preorders which are πp-aligned with allsemivalues.
- Characterizations of specific πp-aligned total preorders (with orwithout the comparison of ordered lists of sets)
Why not to consider probabilistic values?
A probabilistic value πp (or probabilistic power index ) π for thegame v is an n-vector πp(v) = (πp1 (v), πp2 (v), . . . , πpn(v)), suchthat
πpi (v) =∑
S∈2N\{i}
pi (S)(v(S ∪ {i})− v(S)
)(1)
for each i ∈ N and S ∈ 2N\{i}, and p = (pi : 2N\{i} → R+)iinN , isa collection of non negative real functions fulfilling thecondition
∑S∈2N\{i} pi (S) = 1.
Again RESP...
Theorem (R. Lucchetti, S. Moretti, F. Patrone 2012)
Let N be a finite set and let w be a total preorder on 2N . Thenthe following are equivalent:
1. w is aligned w.r.t. all the probabilistic values;
2. w satisfies the RESP property.
Axiom[Double Permutational Responsiveness, DPR]
Order the sets S1,S2, . . . ,Sns+ns−1 in Σsij ∪ Σs−1
ij when you add iand j , respectively:
S1 ∪ {i} w Sl(1) ∪ {j}|⊔
w |⊔
S2 ∪ {i} Sl(2) ∪ {j}|⊔
|⊔
. . . w . . .|⊔
|⊔
Sns+ns−1 ∪ {i} w Sl(ns+ns−1) ∪ {j}
⇔ {i} w {j}
A characterization with possibility of interaction
Theorem (R. Lucchetti, S. Moretti, F. Patrone 2012)
Let N be a finite set and let w be a total preorder on 2N . Thefollowing statements are equivalent:
1) w fulfills the DPR property;
2) w is πp-aligned w.r.t. all the semivalues.
- {1, 2, 3, 4} Ab {2, 3, 4} Ab {3, 4} Ab {4} Ab {3} Ab {2} Ab
{2, 4} Ab {1, 4} Ab {1, 3} Ab {2, 3} Ab {1, 3, 4} Ab {1, 2, 4} Ab
{1, 2, 3} Ab {1, 2} Ab {1} Ab ∅ is πp-aligned for all p, is not PR,but it is DPR.
Next steps
- generalizing: partial orders...
- particularizing: how to represent interaction on specificapplications?
- thinking of the possibility to do a kind a inverse process, notnecessarily respecting the ranking restricted to the singletons.
Thanks!